Question

The outline of a circularly symmetric bottle has a height of 1 metre. The outline first the curve
x^2= (y(1-y)^4+0.1)^2 where x and y are distances measured in metres, k>0 and 0 < or equal to y < or equal to 1.

(a) state the integral used to determine the volume of the bottle.

(b) Use ur calculator or otherwise to evaluate this integral in terms of k

(c) Evaluate k to 2 decimal places given that the capacity of the bottle is 100 litres ( 1 litre = 0.001 m^3)

(d) How wide is the bottle at its widest part?

Answer 1

The graph looks likes this but without the k constant
http://www.wolframalpha.com/input/?i=x^2%3D%28y%281-y%29^4%2B0.1%29^2

Answer 2

I think this question has something to do with volume of revolution
My attempted solution was:
\[\int\limits_{0}^{1} \pi x^2 dy\]
Then i sub in x^2 equation into the integral and tried to evaluate the answer in terms of k.
My answer was a quadratic equation but it consists of many decimal places like for example: 2.325235235k^2 + 0.234234246k + 0.241512352

Answer 3

So from there onwards i m not sure if i did the right thing..

Answer 4

where is k in the equation?

Answer 5

x^2= (ky(1-y)^4+0.1)^2

Answer 6

sry typo error lolz

Answer 7

x^2= (ky(1-y)^4+0.1)^2 << correct equation

Answer 8

my first thought is the lower limit is not 0

Answer 9

oh.. why so..? wouldnt it be rotating about the y axis..?

Answer 10

look at your graph

Answer 11

ok it cuts the y axis at a diff place rite..?

Answer 12

looks like it is slightly negative. wolfram can find the number.

Answer 13

yea but that equation i input in wolfram is without the k constant so will that affect my y intercept..? it does right?

Answer 14

and the graph gaven to me on the paper doesnt show negative values of y

Answer 15

you want the y value when x=0

so if the equation is
0= k*y*(1-y)^4 +0.1
y*(1-y)^4 = -0.1/k
for positive k, and (1-y)^4 positive , y must be negative.

Answer 16

if the equation were
k*( y(1-y)^4 +0.1)=0 (multiplying everything) it does not affect the y value, but y will still be negative

Answer 17

on the other hand, wolfram say y= -0.075, so maybe you can ignore it??

Answer 18

ummz ignore wat?

Answer 19

that the lower limit is not really 0. Just use 0

Answer 20

ok.. but if i used 0 then i integrate it to get my Volume equation in terms of k, i would get a quadratic equation which has lots of decimal places and i tried to solve for k, it came out as k=-2.53 and k=6.53 so i know k > 0 so i take 6.53.. but what is doubting me is that too many decimal places... so shud i just be confident and think "yea its right? "

Answer 21

ok.. assuming i got the k value alrdy then how do i get the widest part? (d)

Answer 22

what did you use for volume?

Answer 23

0.1 m^3

Answer 24

my k value was 2.028*

Answer 25

I can do the problem later on, and post my results. It will take some time though.

Answer 26

no problem :D i ma go sleep first take ur time~ i will check it in like 4 hours time need to go school tmrw

Answer 27

by the way thank you so much for helping me, may god bless you :)

Answer 28

I got k= 2.02816 which to 2 decimals is k= 2.03
max width is 0.532 m